Ring (mathematics)

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In mathematics, a ring is an algebraic structure with two binary operations, commonly called addition and multiplication. These operations are defined so as to emulate and generalize the integers. Other common examples of rings include the ring of polynomials of one variable with real coefficients, or a ring of square matrices of a given dimension.

To qualify as a ring, addition must be commutative and each element must have an inverse under addition: for example, the additive inverse of 3 is -3. However, multiplication in general does not satisfy these properties. A ring in which multiplication is commutative and every element except the additive identity element (0) has a multiplicative inverse (reciprocal) is called a field: for example, the set of rational numbers. (The only ring in which 0 has an inverse is the trivial ring of only one element.)

A ring can have a finite or infinite number of elements. An example of a ring with a finite number of elements is ${\displaystyle \mathbb {Z} _{5}}$, the set of remainders when an integer is divided by 5, i.e. the set {0,1,2,3,4} with operations such as 4 + 4 = 3 because 8 has remainder 3 when divided by 5. A similar ring ${\displaystyle \mathbb {Z} _{n}}$ can be formed for other positive values of ${\displaystyle n}$.

Formal definition

A ring is a set R equipped with two binary operations, which are generally denoted + and · and called addition and multiplication, respectively, such that:

In practice, the symbol · is usually omitted, and multiplication is just denoted by juxtaposition. The usual order of operations is also assumed, so that a + bc is an abbreviation for a + (b·c). The distributive property is specified separately for left and right multiplication to cover cases where multiplication is not commutative, such as a ring of matrices.

Types of rings

Unital ring

A ring in which there is an identity element for multiplication is called a unital ring, unitary ring, or simply ring with identity. The identity element is generally denoted 1. Some authors, notably Bourbaki, demand that their rings should have an identity element, and call rings without an identity pseudorings.

Commutative ring

A ring in which the multiplication operation is commutative is called a commutative ring. Such commutative rings are the basic object of study in commutative algebra, in which rings are generally also assumed to have a unit.

Division ring

A unital ring in which every non-zero element a has an inverse, that is, an element a−1 such that a−1a = aa−1 = 1, is called a division ring or skew field.

Homomorphisms of rings

A ring homomorphism is a mapping ${\displaystyle \pi }$ from a ring ${\displaystyle A}$ to a ring ${\displaystyle B}$ respecting the ring operations. That is,

${\displaystyle \pi (ab)=\pi (a)\pi (b)}$
${\displaystyle \pi (a+b)=\pi (a)+\pi (b)}$

If the rings are unital, it is often assumed that ${\displaystyle \pi }$ maps the identity element of ${\displaystyle A}$ to the identity element of ${\displaystyle B}$.

A homomorphism can map a larger set onto a smaller set; for example, the ring ${\displaystyle A}$ could be the integers ${\displaystyle \mathbb {Z} }$ and could be mapped onto the trivial ring which contains only the single element ${\displaystyle 0}$.

Subrings

If ${\displaystyle A}$ is a ring, a subset ${\displaystyle B}$ of ${\displaystyle A}$ is called a subring if ${\displaystyle B}$ is a ring under the ring operations inherited from ${\displaystyle A}$. It can be seen that this is equivalent to requiring that ${\displaystyle B}$ be closed under multiplication and subtraction.

If ${\displaystyle A}$ is unital, some authors demand that a subring of ${\displaystyle A}$ should contain the unit of ${\displaystyle A}$.

Ideals

A two-sided ideal of a ring ${\displaystyle A}$ is a subring ${\displaystyle I}$ such that for any element ${\displaystyle a}$ in ${\displaystyle A}$ and any element ${\displaystyle b}$ in ${\displaystyle I}$ we have that ${\displaystyle ab}$ and ${\displaystyle ba}$ are elements of ${\displaystyle I}$. The concept of ideal of a ring corresponds to the concept of normal subgroups of a group. Thus, we may introduce an equivalence relation on ${\displaystyle A}$ by declaring that two elements of ${\displaystyle A}$ are equivalent if their difference is an element of ${\displaystyle I}$. The set of equivalence classes is then denoted by ${\displaystyle A/I}$ and is a ring with the induced operations.

If ${\displaystyle h:A\rightarrow B}$ is a ring homomorphism, then the kernel of h, defined as the inverse image of 0, ${\displaystyle \{x\in A:h(x)=0\}}$, is an ideal of ${\displaystyle A}$. Conversely, if ${\displaystyle I}$ is an ideal of ${\displaystyle A}$, then there is a natural ring homomorphism, the quotient homomorphism, from ${\displaystyle A}$ to ${\displaystyle A/I}$ such that ${\displaystyle I}$ is the set of all elements mapped to 0 in ${\displaystyle A/I}$.

Examples

• The trivial ring {0} consists of only one element, which serves as both additive and multiplicative identity.
• The integers form a ring with addition and multiplication defined as usual. This is a commutative ring.
• The set of polynomials forms a commutative ring.
• The set of square ${\displaystyle n\times n}$ matrices forms a ring under componentwise addition and matrix multiplication. This ring is not commutative if n>1.
• The set of all continuous real-valued functions defined on the interval [a,b] forms a ring under pointwise addition and multiplication.

Constructing new rings from given ones

• For every ring ${\displaystyle R}$ we can define the opposite ring ${\displaystyle R^{op}}$ by reversing the multiplication in ${\displaystyle R}$. Given the multiplication ${\displaystyle \cdot }$ in ${\displaystyle R}$, the multiplication ${\displaystyle \star }$ in ${\displaystyle R^{op}}$ is defined as ${\displaystyle a\star b:=b\cdot a}$. The "identity map" from ${\displaystyle R}$ to ${\displaystyle R^{op}}$, mapping each element to itself, is an isomorphism if and only if ${\displaystyle R}$ is commutative. However, even if ${\displaystyle R}$ is not commutative, it is still possible for ${\displaystyle R}$ and ${\displaystyle R^{op}}$ to be isomorphic using a different map. For example, if ${\displaystyle R}$ is the ring of ${\displaystyle n\times n}$ matrices of real numbers, then the transposition map from ${\displaystyle R}$ to ${\displaystyle R^{op}}$, mapping each matrix to its transpose, is an isomorphism.
• The center of a ring ${\displaystyle R}$ is the set of elements of ${\displaystyle R}$ that commute with every element of ${\displaystyle R}$; that is, ${\displaystyle c}$ is an element of the center if ${\displaystyle cr=rc}$ for every ${\displaystyle r\in R}$. The center is a subring of ${\displaystyle R}$. We say that a subring ${\displaystyle S}$ of ${\displaystyle R}$ is central if it is a subring of the center of ${\displaystyle R}$.
• The direct product of two rings R and S is the cartesian product R×S together with the operations
(r1, s1) + (r2, s2) = (r1+r2, s1+s2) and
(r1, s1)(r2, s2) = (r1r2, s1s2).
With these operations R×S is a ring.
• More generally, for any index set J and collection of rings ${\displaystyle \{R_{j}\}_{j\in J}}$, the direct product and direct sum exist.
• The direct product is the collection of "infinite-tuples" ${\displaystyle \{r_{j}\}_{j\in J}}$ with component-wise addition and multiplication as operations.
• The direct sum of a collection of rings ${\displaystyle \{R_{j}\}_{j\in J}}$ is the subring of the direct product consisting of all infinite-tuples ${\displaystyle \{r_{j}\}_{j\in J}}$ with the property that rj=0 for all but finitely many j. In particular, if J is finite, then the direct sum and the direct product are isomorphic, but in general they have quite different properties.
• Since any ring is both a left and right module over itself, it is possible to construct the tensor product of R over a ring S with another ring T to get another ring, provided S is a central subring of R and T.

History

The study of rings originated from the study of polynomial rings and algebraic number fields in the second half of the nineteenth century, amongst other by Richard Dedekind. The term ring itself, however, was coined by David Hilbert in 1897.

• Special types of rings:
• Constructions of rings

References

Fraleigh, John B. 2003. A First Course in Abstract Algebra. 7th ed. Boston: Addison-Wesley

Hilbert, David. 1897. Die Theorie der algebraische Zahlkoerper, ahresbericht der Deutschen Mathematiker Vereiningung vol. 4.

Lang, Serge. 2002. Algebra. 3rd ed. New York: Springer